Hostname: page-component-586b7cd67f-2brh9 Total loading time: 0 Render date: 2024-11-23T21:27:00.979Z Has data issue: false hasContentIssue false
  • English
  • Français

NAIVE TRUTH AND RESTRICTED QUANTIFICATION: SAVING TRUTH A WHOLE LOT BETTER

Published online by Cambridge University Press:  14 October 2013

HARTRY FIELD
HARTRY FIELD*
Affiliation:
New York University and University of Birmingham
*
*DEPARTMENT OF PHILOSOPHY, NEW YORK UNIVERSITY 5 WASHINGTON PLACE, NEW YORK NY 10003, USA and DEPARTMENT OF PHILOSOPHY, UNIVERSITY OF BIRMINGHAM EDGBASTON, BIRMINGHAM, B15 2TT UK E-mail: [email protected]
Get access Rights & Permissions [Opens in a new window]

Abstract

Restricted quantification poses a serious and under-appreciated challenge for nonclassical approaches to both vagueness and the semantic paradoxes. It is tempting to explain “All A are B” as “For all x, if x is A then x is B”; but in the nonclassical logics typically used in dealing with vagueness and the semantic paradoxes (even those where ‘if … then’ is a special conditional not definable in terms of negation and disjunction or conjunction), this definition of restricted quantification fails to deliver important principles of restricted quantification that we’d expect. If we’re going to use a nonclassical logic, we need one that handles restricted quantification better.

The challenge is especially acute for naive theories of truth—roughly, theories that take True(〈A〉) to be intersubstitutable with A, even when A is a “paradoxical” sentence such as a Liar-sentence. A naive truth theory inevitably involves a somewhat nonclassical logic; the challenge is to get a logic that’s compatible with naive truth and also validates intuitively obvious claims involving restricted quantification (for instance, “If S is a truth stated by Jones, and every truth stated by Jones was also stated by Smith, then S is a truth stated by Smith”). No extant naive truth theory even comes close to meeting this challenge, including the theory I put forth in Saving Truth from Paradox. After reviewing the motivations for naive truth, and elaborating on some of the problems posed by restricted quantification, I will show how to do better. (I take the resulting logic to be appropriate for vagueness too, though that goes beyond the present paper.)

In showing that the resulting logic is adequate to naive truth, I will employ a somewhat novel fixed point construction that may prove useful in other contexts.

Type
Research Article
Information
The Review of Symbolic Logic , Volume 7 , Issue 1 , March 2014 , pp. 147 - 191
Copyright
Copyright © Association for Symbolic Logic 2013 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

BIBLIOGRAPHY

Anderson, A. R., & Belnap, N. (1975). Entailment, Vol. 1. Princeton, NJ: Princeton University Press.Google Scholar
Bacon, A. (2013). A new conditional for naive truth theory. Notre Dame Journal of Formal Logic, 54, 87–103.CrossRefGoogle Scholar
Beall, J. C. (2009). Spandrels of Truth. Oxford, UK: Oxford University Press.CrossRefGoogle Scholar
Beall, J. C., Brady, R., Hazen, A., Priest, G., & Restall, G. (2006). Relevant restricted quantification. Journal of Philosophical Logic, 35, 303–324.CrossRefGoogle Scholar
Brady, R. (2006). Universal Logic. Stanford, CA: CSLI Publications.Google Scholar
Field, H. (2003). No fact of the matter. Australasian Journal of Philosophy, 81, 457–480.CrossRefGoogle Scholar
Field, H. (2004). The semantic paradoxes and the paradoxes of vagueness. In Beall, J. C., editor. Liars and Heaps. Oxford, UK: Oxford University Press, pp. 262–311.CrossRefGoogle Scholar
Field, H. (2008). Saving Truth from Paradox. Oxford, UK: Oxford University Press.CrossRefGoogle Scholar
Gupta, A., & Belnap, N. (2003). The Revision Theory of Truth. Cambridge, MA: MIT Press.Google Scholar
Hajek, P., Paris, J., & Sheperdson, J. (2000). The liar paradox and fuzzy logic. Journal of Symbolic Logic, 65, 339–346.CrossRefGoogle Scholar
Halbach, V. (2011). Axiomatic Theories of Truth. Cambridge, UK: Cambridge University Press.CrossRefGoogle Scholar
Horsten, L. (2011). The Tarskian Turn: Deflationism and Axiomatic Truth. Cambridge, MA: MIT Press.CrossRefGoogle Scholar
Priest, G. (2008). An Introduction to Non-classical Logic (second edition). Cambridge, UK: Cambridge University Press.CrossRefGoogle Scholar
Restall, G. (1992). Arithmetic and truth in Lukasiewicz’s infinitely-valued logic. Logique et Analyse, 139–140, 303–312.Google Scholar
White, R. (1979). The consistency of the axiom of comprehension in the infinite-valued predicate logic of Lukasiewicz. Journal of Philosophical Logic, 8, 509–534.CrossRefGoogle Scholar